In this paper, we suggest approximate algorithms for the reconstruction of sparse high-dimensional trigonometric polynomials, where the support in frequency domain is unknown. Based on ideas of constructing rank-1 lattices component-by-component, we adaptively construct the index set of frequencies belonging to the non-zero Fourier coefficients in a dimension incremental way. When we restrict the search space in frequency domain to a full grid [ − N , N ] d ∩ Z d of refinement N ∈ N and assume that the cardinality of the support of the trigonometric polynomial in frequency domain is bounded by the sparsity s ∈ N , our method requires O ( d s 2 N ) samples and O ( d s 3 + d s 2 N log ( s N ) ) arithmetic operations in the case N ≲ s ≲ N d . Moreover, we discuss possibilities to reduce the number of samples and arithmetic operations by applying methods from compressed sensing and a version of Prony's method. For the latter, the number of samples is reduced to O ( d s + d N ) and the number of arithmetic operations is O ( d s 3 ) . Various numerical examples demonstrate the efficiency of the suggested method.